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The shape distribution of boulders on Asteroid 25143 Itokawa: Comparison with fragments from impact experiments
Icarus 207 (2010) 277–284
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The shape distribution of boulders on Asteroid 25143 Itokawa: Comparison with fragments from impact experiments Tatsuhiro Michikami a,b,*, Akiko M. Nakamura c, Naru Hirata d a
Fukushima National College of Technology, Iwaki, Fukushima 970-8034, Japan Planetary and Space Sciences Research Institute, The Open University, Walton Hall, Milton Keynes, MK7 6AA, United Kingdom c Graduate School of Science/Center for Planetary Science, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan d Department of Computer Science and Engineering/ARC-Space, CAIST, The University of Aizu, Ikki-machi, Aizu-Wakamatsu, Fukushima 965-8580, Japan b
a r t i c l e
i n f o
Article history: Received 10 June 2009 Revised 30 September 2009 Accepted 4 October 2009 Available online 22 October 2009 Keywords: Asteroids Asteroid Itokawa Asteroid Eros Asteroids, Surfaces Regoliths
a b s t r a c t Laboratory impact experiments have found that the shape of fragments over a broad size range is distribp uted around the mean value of the axial ratio 2: 2:1, which is independent of a wide range of experimental conditions. We report the shape statistics of boulders with size of 0.1–30 m on the surface of Asteroid 25143 Itokawa based on high-resolution images obtained by the Hayabusa spacecraft in order to investigate whether their shape distribution is similar to the distribution obtained for fragments (smaller than 0.1 m) in laboratory impact experiments. We also investigated the shapes of boulders with size of 0.1–150 m on Asteroid 433 Eros using a few arbitrary selected images by the NEAR spacecraft, in order to compare those with the shapes on Asteroid Itokawa. In addition, the shapes of small- and fastrotating asteroids (diameter <200 m and rotation period <1 h), which are natural fragments from past impact events among asteroids, were inferred from archived light curve data taken by ground-based telescopes. The results show that the shape distributions of laboratory fragments are similar to those of the boulders on Eros and of the small- and fast-rotating asteroids, but are different from those on Itokawa. However, we propose that the apparent difference between the boulders of Itokawa and the laboratory fragments is due to the migration of boulders. Therefore, we suggest that the shape distributions of the boulders ranging from 0.1 to 150 m in size and the small- and fast-rotating asteroids are similar to those obtained for the fragments generated in laboratory impact experiments. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Laboratory impact experiments have found that the shapes of fragments as defined by axes a, b and c, these being the maximum dimensions of the fragment in three mutually orthogonal planes (a P b P c) are distributed around mean values of the axial ratios b/a 0.7 and c/a 0.5, i.e., corresponding to a:b:c in the simple p proportion 2: 2:1. This result indicates a general property of collisional fragments which is repeated with great regularity in widely different experimental conditions such as projectile velocity, target shape, composition and strength (Fujiwara et al., 1978; Matsui et al., 1982, 1984; Bianchi et al., 1984; Capaccioni et al., 1984, 1986). In general, the shape distributions of small asteroids less than tens of kilometers in diameter are considered to be similar to distributions obtained for fragments generated in laboratory impact experiments (Fujiwara et al., 1978; Capaccioni et al., 1984, 1986). The shapes of most asteroids can be inferred from the ob-
* Corresponding author. Address: Fukushima National College of Technology, Iwaki, Fukushima 970-8034, Japan. E-mail address: [email protected] (T. Michikami). 0019-1035/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2009.10.008
served light curve amplitude. The observed amplitudes imply that the shape distribution of small asteroids is similar to those of laboratory fragments although the observed small asteroids tend to have slightly more elongated shapes (e.g., Catullo et al., 1984; Binzel et al., 1989). Most asteroids are fragments produced as a result of the collisional disruption of parent bodies, or are gravitationally bound rubble piles of fragments. The small asteroids have a wide range of shape elongations, while the shapes of large asteroids are more spherical due to self-gravitational effects (e.g., Capaccioni et al., 1984). The sizes of the observed laboratory fragments are less than 101 m, and the sizes of the observed asteroids are larger than 102 m. In previous studies, there has been no research on the shape distribution of the fragments with size of between 101 and 102 m. In the last few years, the high-resolution images taken by the NEAR and Hayabusa spacecraft have shown that there are many boulders on Asteroids Eros and Itokawa, respectively (e.g., Thomas et al., 2001; Fujiwara et al., 2006). The sizes of these boulders range from tens of centimeters to several tens of meters, occasionally as large as 150 meters. The boulders are formed by impact cratering or catastrophic disruption of the parent asteroid. The discovery of many
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boulders provides the opportunity for the statistical estimation of the shapes of the fragments in space. We report the shape statistics of boulders on the surface of Itokawa based on high-resolution images obtained by the Hayabusa spacecraft. Then we investigate, though limited to only a very small sampling, the boulder shapes on Asteroid Eros in order to compare those with the shapes of boulders on Asteroid Itokawa. In addition, in order to confirm the shapes of small asteroids as fragments with size less than 200 m, the shapes of fast-spinning asteroids (with spin period less than 1 h) are inferred from the archived light curve observation data (Harris and Pravec; EAR-A5-DDR-DERIVED-LIGHTCURVE-V10.0. ‘‘http://www.psi.edu/pds/ resource/lc.html”). In the last few years, smaller (less than 200 m), fast-rotating asteroids have been discovered and these are considered to be monolithic bodies (not rubble piles) generated by impact cratering or catastrophic disruption of the parent asteroids (Holsapple, 2007). Based on these analyses, we investigate whether the shape distributions of these fragments found in space ranging from tens of centimeters to a few hundred meters are similar to the distribution obtained for fragments less than 10 cm in diameter generated in laboratory impact experiments. 2. Methodology We define isolated positive relief features on the surface of an asteroid as boulders. Isolated positive relief features might include partially buried boulders, weathered boulders with subdued outlines, piles of regolithic scree, protruding bedrocks, raised crater rims and intersecting crater walls. From the images taken by spacecraft, it is difficult to distinguish clearly between these features, especially for small positive relief. For large positive relief features (>5 m), we took out only what is considered to be a boulder by comparing a few hundred images with stereo coverage acquired from the nominal hovering position (Home position) at an altitude of about 7 km. Almost all large positive relief features would be boulders. The shape statistics of the large boulders (>5 m) on the entire surface of Itokawa were investigated from eight images (resolutions 0.4 m/pixel) acquired on 19–26th October 2005. In this study, we call the mean size of a boulder larger than 5 m ‘‘a large boulder”. This critical value is not hampered by the limited spatial resolution on the entire asteroid imaged by the spacecraft. When we measured the size of a boulder, we chose a best image in which the outline of the boulder is clear, the emission angle between surface normal and spacecraft position is the smallest and the apparent dimension is the longest in the eight images. We started by measuring the apparent longest dimension of the boulder (a0 ) and then the apparent longest dimension (b0 ) perpendicular to this. We made corrections in the axial ratios of the large boulders on Itokawa because the apparent dimension of the boulder might be somewhat affected by the emission angle (data excluding the large boulders on Itokawa were not corrected because the emission angles of these data are small). In our study, as a first approximation, the correction value of the axial ratio is calculated using the following equation: 0
b b ¼ a a0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sin j þ cos2 j cos2 e 2
sin j cos2 e þ cos2 j
;
ð1Þ
where a and b are the correction value of a0 and b0 , respectively, and e and j are the emission angle and the angle between the asteroid surface projection axis of the emission vector and the a0 axis, respectively. This equation can be obtained geometrically assuming that the a0 and b0 are the projected axes of the a and b of the boulder on the plane with e and j (note that, in this equation, we do not take into account of the ellipsoid or the true shape of the boulder but
treat it as two axes a and b). If the direction of the a0 axis is the same as the asteroid surface projection axis of the emission vector (j = 0°), b0 equals b and a0 becomes a(cos e), eventually the correction value of the axial ratio (b/a) becomes (b0 /a0 )(cos e). If the direction of the a0 axis is perpendicular to the asteroid surface projection axis of the emission vector (j = 90°), a0 equals a and b0 becomes b(cos e), eventually the correction value of the axial ratio (b/a) becomes (b0 /a0 )(1/ cos e). When we make corrections, the correction values of the axial ratios in small fraction of boulders become over 1; the dimension of the a axis becomes smaller than that of the b axis. In this case, we excluded these data because the longest dimension cannot be measured correctly. Note that Eq. (1) is not applicable when the orientations of the apparent longest axis are not parallel to the asteroid surface. However, in our observation, because the orientations of the apparent dimension axis of most large boulders are parallel to the asteroid surface, Eq. (1) would be applicable in statistical viewpoint. The dimensions of a and b might not represent the actual longest and middle axes of the boulder because these are apparent dimensions of the boulder on the asteroid surface, which might be partially buried. The shape statistics of small boulders with size less than 5 m were derived from several smaller, representative regions’ images; these were estimated from six close-up images acquired on 9–12th November 2005 (Fig. 1). We counted all positive relief features in these six images and defined these as boulders. Boulder mapping is affected by variations in spatial resolution and in the availability of stereo coverage. The regions at middle and low latitudes were viewed from several images at resolution 0.4 m/pixel and with substantial stereo coverage. In contrast, the regions at high latitude were viewed from only a few images at resolution 0.4 m/pixel and with no stereo coverage. Therefore, it is possible that several boulders located at high latitude were overlooked in our counting.
3. Observation results of the boulders on Itokawa There are 373 boulders larger than 5 m on the entire surface of Itokawa, using the method of Michikami et al. (2008). The numbers of boulders larger than 10 m is about 40. The largest boulder is called ‘‘Yoshinodai” with dimensions of 50 m by 30 m by 20 m (Fujiwara et al., 2006). The shape distributions of large boulders on the surface of Itokawa and of the fragments in laboratory experiments (Capaccioni et al., 1984) are shown in Figs. 2 and 3, respectively. Note that the mean value of the apparent axial ratio (b/a) of the boulders larger than 5 m on the entire surface of Itokawa is 0.62 (Table 1), which is smaller than that (0.72) of the fragments in laboratory experiments. Both of rank-sum test and t test show that the difference between the observed large boulders and laboratory fragments is statistically significant at the 95% confidence level. The typical large boulders of Itokawa appear to have somewhat more elongated shapes compared with laboratory fragments. The apparent axial ratios of the large boulders are distributed with a gentle peak compared with those of the fragments in the laboratory experiments; these boulders have various shapes such as spheres and elongated ellipses. In particular, there are no fragments with b/a < 0.3 in laboratory experiments, while there are several large boulders with b/a < 0.3 on Itokawa. The mean value of the apparent axial ratio of the boulders larger than 10 m is also similar to that of the boulders larger than 5 m (Table 1). We also investigated the shape distributions of the large boulders among different areas of Itokawa, such as the east and west sides, the rough and smooth areas, and the ‘‘head” and ‘‘body” portions. The surface of Itokawa is separated into ‘‘rough area”, which occupies about 80% of Itokawa’s surface with numerous boulders,
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Fig. 1. (a) East side of Itokawa. (b–g) Close-up images acquired on 9–12th November 2005. Scale bars are indicated on the top right of each image. Image IDs: (a) ST2417413276, (b) ST2539437177, (c) ST2539429953, (d) ST2539423137, (e) ST2532629277, (f) ST2539451609 and (g) ST2539444467. The locations of the regions covered by images (b–g) are shown in (a).
and ‘‘smooth area” covered by finer particles. A surprising number of positive relief features appear on the rough area. The shape of Itokawa resembles a sea otter, with a smaller, rather round head and a larger oval body mass. The results show that the mean values of the apparent axial ratios among these areas are similar (Table 1). As an exception, the mean value (0.56) of the apparent axial ratio on the smooth area is smaller than those of other areas. The difference might be due to the small number of large boulders on the smooth area. As mentioned before, the shape distributions of the boulders less than 5 m were estimated from six close-up images acquired on 9–12th November 2005. These images were sampled from the boundary of MUSES-C Regio and the rough area of the larger body moving from the west to the east along the equator of the east side (Fig. 1). The altitude of the spacecraft ranged from 60 to 646 m. Fig. 4 shows the shape distributions of the boulders ranging from 0.1 to 5 m in six close-up images. The counted numbers of the
boulders are several hundreds for each image. Image (e) in Fig. 1 covers a wide area and contains three smaller areas of images (b–d). Note that the mean values of the apparent axial ratio (b/a) of all of the boulders with size of 0.1–5 m is 0.68 (Table 2), which is slightly smaller than that (0.72) of the fragments in laboratory experiments. Although the difference between these looks small, it is statistically significant at the 95% confidence level (both of rank-sum test and t test). The small boulders of Itokawa also appear to have slightly more elongated shapes compared with those of laboratory fragments. The mean values of the apparent axial ratio for each image appear to be roughly the same (Table 2). The peaks of the shape distributions differ slightly for each image (Fig. 4). The shape distributions of the boulders on images (d) and (f) are peaked at b/a 0.6, while other shape distributions are peaked at b/a 0.7. The mean values (0.68) of the apparent axial ratio of all of the boulders with size of 0.1–5 m is larger than that (0.62) of the boul-
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Fig. 2. The shape distributions of boulders larger than 5 m on the surface of Itokawa. The horizontal axis shows the apparent axial ratios (b/a) of the boulders. The vertical axis indicates the number of boulders. We made corrections in the axial ratios of the large boulders on Itokawa using Eq. (1). The total shape distributions with the correction and without the correction are shown on the top left and the top right, respectively. The total number of the large boulders is 373 and the total number of those with the correction value is 330. The mean value of the apparent axial ratio (b/a) of the large boulders is 0.62, which is similar to that without the correction (0.61). The surface is divided into different areas, such as the east and west sides, the rough and smooth areas, and the head and body portions. The mean values of apparent axial ratios in each area are given in Table 1. Data were collected from the eight images of ST2473604354, ST2481211873, ST2482160259, ST2484352917, ST2485860275, ST2492225173, ST2492513077, and ST2493031594 at an altitude between 3.779 and 4.913 km.
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and 0156087736). The mean apparent axial ratio of these boulders with size of 4–17 m is 0.72 (the counted number is 41), and this value is also similar to those of laboratory fragments. The calculations of rank-sum test and t test show that these distributions are also similar. Still smaller boulders with size of 0.1–3.8 m are assessed from final images of the landing site on Eros acquired by NEAR Shoemaker on February 12, 2001 (Image IDs 0157416873, 0157417133 and 0157417198). The total counted number is 163. The apparent mean axial ratio of small boulders with size of 0.1– 3.8 m on the close-up images of Eros is 0.73, and this value is also similar to those of laboratory fragments. The calculations of ranksum test and t test show that these distributions are also similar. Therefore, the shape distribution of the boulders in the wide ranges of sizes on Eros is thought to be similar to those of laboratory fragments, although these data are derived from a small number of sample images. Fig. 3. The shape distribution of fragments in laboratory experiments (Fig. 2B in Capaccioni et al. (1984)). The mean value of the axial ratio (b/a) of the fragments is 0.72.
Table 1 The mean values of axial ratios (b/a) of boulders larger than 5 m in each area of Itokawa. These values are the correction ones using Eq. (1).
Total (without correction) Total East West Head Body Rough Smooth >10 m
Mean
Standard deviation
Number of boulders
0.61
0.19
373
0.62 0.61 0.64 0.62 0.62 0.63 0.56 0.63
0.19 0.18 0.20 0.20 0.18 0.19 0.12 0.19
330 166 164 101 229 307 23 38
ders larger than 5 m on the entire surface of Itokawa (Table 2). The small boulders less than 5 m appear to have less elongated shapes compared with the boulders larger than 5 m on the entire surface. In image (e) which covers a wide area, the mean values of the axial ratio (b/a) of the boulders with size of 1–5 m and with size of 2– 5 m are 0.66 and 0.60, respectively. This means that the mean values of the apparent axial ratio (b/a) of the boulders decrease with increasing the size; as the size increases, it looks like the boulders’ shape is getting more elongated for the size range of 0.1–5 m. 4. Shape distribution of boulders on Eros In order to compare with the shape of the boulders on Itokawa, preliminary measurements of shape of the boulders on Eros have been made in a few arbitrary selected locations from six images acquired by the NEAR Shoemaker spacecraft. The shapes of the boulders on Eros were measured using the same method for investigating the shape of the boulders on Itokawa; the shape distributions are shown in Fig. 5. Firstly, the population of large boulders with size of 60–150 m on Eros is sampled from the image taken during December 25, 2000 (Image ID 0153130598). The image shows inside the rim of Eros saddle region and the orbital altitude is 38 km (resolution 3 m/pixel). The mean apparent axial ratio of large boulders with size of 60–150 m is 0.73 (the counted number is 20), and this value is similar to those of laboratory fragments. The calculations of rank-sum test and t test show that these distributions are similar. Secondly, the shapes of relatively smallsized boulders with size of 4–17 m are derived from two low-altitude images in January 26 and 28, 2001 (Image IDs 0155888661
5. Shape distribution of small- and fast-rotating asteroids According to Binzel et al. (1989), the observed mean amplitude of light curve of asteroids is slightly higher than expected for the shape distribution of laboratory fragments; the observed asteroids tend to have slightly more elongated shapes. There is as yet no clear evidence to show that the shape distribution of small asteroids is the same as those of fragments in laboratory impact experiments. Nonellipsoidal shapes, scattering properties of the surface, shadowing and albedo effects at nonzero phase angles may be enhancing the observed amplitudes. Space missions have visited only a few asteroids and shapes for these have been determined accurately (e.g., Fujiwara et al., 2006). Radar observations have revealed shapes of Near-Earth Objects encountering the Earth (e.g., Ostro et al., 2007), while high-resolution imaging is restricted to the largest asteroids (Tanga et al., 2003; Marchis et al., 2006). In order to confirm the shapes of small asteroids as fragments with size less than 200 m, we researched the shapes of the fastrotating (rotation period less than 1 h) asteroids. Fast-rotating asteroids have been discovered in the last few years. They have diameters of less than 200 m, and are all Near Earth Asteroids. These spin speeds clearly exceed the rubble-pile spin limits (Holsapple, 2007). Thus, these small- and fast-rotating asteroids must have some tensile strength and are considered to be monolithic bodies generated by impact cratering or catastrophic disruption of the parent asteroids (Holsapple, 2007). Therefore, it is likely that these small asteroids have the original shapes of the fragments produced by impact phenomena. Light curve amplitudes (defined as the peak-to-peak variation) provide a crude indicator for asteroid shapes. The compilation of light curve amplitude data is acquired from the asteroid complication of Harris and Pravec (EAR-A-5-DDR-DERIVED-LIGHTCURVEV10.0). According to Binzel et al. (1989), for a triaxial ellipsoid rotating about the c axis, the light curve amplitude is given by
AðhÞ ¼ 2:5 log
a b
2
1:25 log
a2 cos2 h þ c2 sin h 2
2
!
b cos2 h þ c2 sin h
;
ð2Þ
where h is the polar aspect viewing angle (the angle between the line of sight and the axis of rotation). If an asteroid is viewed pole-on (h = 0°), then no change in projected surface area is seen and the expected amplitude is zero. If an asteroid is viewed at an equatorial aspect (h = 90°) such that the rotation axis is perpendicular to the line of sight, then the second term in the above equation is zero and the maximum amplitude would be expected as the projected surface area varies from pac to pbc. The polar aspect viewing angle is unknown for an asteroid observed during only one opposition (Magnusson et al., 1989). If we assume an isotropic distribution of asteroid spin vectors, then
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Fig. 4. The shape distributions of boulders ranging from 0.1 to 5 m on the six close-up images (corresponding to Fig. 1). The mean values of apparent axial ratios in each image are given in Table 2.
T. Michikami et al. / Icarus 207 (2010) 277–284 Table 2 The mean values of axial ratios (b/a) of boulders ranging from 0.1 to 5 m in each close-up images of Itokawa. No.
Mean
Standard deviation
Number of boulders
Size range (m)
b c d e
0.67 0.67 0.66 0.68 0.66 0.60 0.69 0.69 0.68
0.16 0.17 0.16 0.16 0.17 0.17 0.16 0.15 0.17
276 410 439 495 309 53 173 240 2033
0.12–4.65 0.22–3.84 0.23–2.74 0.58–5.37 1–5.37 2–5.37 0.14–2.15 0.05–2.56 0.05–5.37
f g Total
the mean value of the aspect angles observed during one opposition will be 60° (Binzel et al., 1989). However, a recent study of the Koronis family shows that the distribution of asteroid spin vectors is not isotropic (Slivan, 2002). It is possible that the rotation axis of some asteroids can become perpendicular to the asteroid’s orbit plane. In this study, we calculated the axial ratio of small- and fast-rotating asteroids from the amplitudes of these asteroids using the value of h = 90°, for simplicity. If the distribution of the asteroid spin vectors is isotropic, then the axial ratio becomes small compared with our estimated value. The shape distribution of 42 small- and fast-rotating asteroids (diameter <200 m and rotation period <1 h) is shown in Fig. 6. Note that the mean axial ratio is 0.71, which is similar to those of laboratory fragments. The calculations of rank-sum test and t test show that these distributions are similar.
283
6. Discussion The shape distributions of the boulders with size of 0.1–150 m on Eros and the small- and fast-rotating asteroids (diameter <200 m and rotation period <1 h) are similar to those of fragments generated in laboratory impact experiments. On the other hand, the shape distributions of boulders on Itokawa are different from those of laboratory fragments; the boulders on Itokawa appear to have more elongated shapes. However, the actual shape distribution of the boulders on Itokawa would be similar to those of laboratory fragments. The apparent difference between the boulders of Itokawa and the laboratory fragments would be due to the migration of boulders (regolith migration). The regolith migration indicated by Miyamoto et al. (2007) brings about the different sedimentation of the boulders between Itokawa and Eros. The explanation of regolith migration on Itokawa is as follows. Most of boulders on Itokawa originated from the disruption of a larger parent body, and then these boulders were uniformly spread over the entire surface (Miyamoto et al., 2007; Michikami et al., 2008). The smaller boulders (gravels) were redistributed after their accumulation by global vibration (seismic shaking) which was caused by impact cratering, etc. Finer particles such as gravel migrated into the smooth terrains which are areas of low gravitational potential. This is because finer particles have higher mobility due to their lower friction angle. On the other hand, the larger boulders could not move easily and some large boulders were stranded at the surface owing to their large friction angle and then form rough terrains. These mechanisms are explained in detail by Miyamoto et al. (2007) as granular pro-
Fig. 5. The shape distributions of the boulders of size 0.1–150 m on Eros, which are determined for a few arbitrary selected locations from six images acquired by NEAR Shoemaker. The mean apparent values of axial ratios are 0.73 (for size 60–150 m), 0.72 (for size 4–17 m) and 0.73 (for size 0.1–3.8 m).
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tory-scale collision experiments. However, we believe that the possibility of this is low, because the shape distribution of boulders in the same size range on Eros is similar to laboratory fragments. Finally, we suggest that the actual shape distributions of the boulders on Itokawa are similar to those of laboratory fragments. Based on our analyses, we conclude that the shape distributions of the boulders ranging from 0.1 to 150 m in size and the smalland fast-rotating asteroids (diameter <200 m and rotation period <1 h) are similar to that obtained for fragments generated in laboratory impact experiments. Acknowledgments We would like to thank all members of Hayabusa mission team for their support of the data acquisition. We thank S. Green and A. Hagermann for useful discussion. Fig. 6. The shape distribution of the small- and fast-rotating asteroids (diameter <200 m and rotation period <1 h). The mean value of axial rations is 0.71.
References
cesses (granular processes become major resurfacing processes because of Itokawa’s small size). The c axis of the gravels gradually would become perpendicular to the surface of Itokawa during the granular process, owing to their lower friction angle and gravitational stability. As a result, the apparent shape distributions of small boulders (gravels) tend to approach those of laboratory fragments as the size of small boulders decreases: For boulders with sizes ranging from 2 to 5 m, 1 to 5 m, and 0.1 to 5 m, the mean apparent axial ratios are 0.60, 0.66, and 0.68, respectively. On the other hand, the orientations of the c axis of some large boulders may not be perpendicular to the surface because it is likely that some large boulders are stranded at the surface and remain at their original orientations owing to their large friction angle (note that, in our observation, the orientations of the a axis of most large boulders are parallel to the asteroid surface. Only the orientations of the c axis (or the b axis) would be somewhat random). Consequently, the appearance of the large boulders tends to more elongated shapes: the apparent axial ratios of the boulders larger than 5 m is 0.62. As one piece of evidence, the very large boulder near the neck between head and body portions is taller than it is wide, which is a rather unstable orientation. On Eros, almost all boulders would be reoriented after their accumulation by global vibration (e.g., Richardson et al., 2005) because the gravity of Eros is considerably larger than that of Itokawa. Therefore, the c axis of all boulders are thought to become perpendicular to the surface, and then the shape distribution of the boulders on Eros is similar to those of laboratory fragments: the mean apparent axial ratio of the boulders on Eros is about 0.73. There is a possibility that the approximation of Eq. (1) is not enough to make corrections in the axial ratio of the boulders. However, the apparent mean axial ratio of the boulders which were observed at the emission angles near 0° is 0.63 (the counted number 32), which is similar to that of the total boulders over the entire surface of Itokawa. Therefore, the approximation of Eq. (1) is considered to be enough to make corrections in the axial ratio of the boulders. Note that we do not rule out the possibility that the boulders on Itokawa are actually more elongated than predicted by the labora-
Bianchi, R., Capaccioni, F., Cerroni, P., Coradini, M., Flamini, E., Hurren, P., Martelli, G., Smith, P.N., 1984. Experimental simulation of asteroidal fragmentation by macroscopic hypervelocity impacts against free falling bodies. Astron. Astrophys. 139, 1–6. Binzel, R.P., Farinella, P., Zappala, V., Cellino, A., 1989. Asteroid Rotation Rates: Distributions and Statistics. Asteroid II. University of Arizona Press, Tucson. pp. 416–441. Capaccioni, F., Cerroni, P., Coradini, M., Farinella, P., Flamini, E., Martelli, G., Paolicchi, P., Smith, P.N., Zappala, V., 1984. Shapes of asteroids compared with fragments from hypervelocity impact experiments. Nature 308, 832–834. Capaccioni, F., and 10 colleagues, 1986. Asteroidal catastrophic collisions simulated by hypervelocity impact experiments. Icarus 66, 487–514. Catullo, V., Zappala, V., Farinella, P., Paolicchi, P., 1984. Analysis of the shape distribution of asteroids. Astron. Astrophys. 138, 464–468. Fujiwara, A., Kamimoto, G., Tsukamoto, A., 1978. Expected shape distribution of asteroids obtained from laboratory impact experiments. Nature 272, 602– 603. Fujiwara, A., and 21 colleagues, 2006. The rubble-pile Asteroid Itokawa as observed by Hayabusa. Science 312, 1330–1334. Holsapple, K.A., 2007. Spin limits of Solar System bodies: From the small fastrotators to 2003 EL61. Icarus 187, 500–509. Magnusson, P., Barucci, M.A., Drummond, J.D., Lumme, K., Ostro, S.J., 1989. Determination of Pole Orientations and Shapes of Asteroids. Asteroid II. University of Arizona Press, Tucson. pp. 67–97. Marchis, F., Kaasalainen, M., Hom, E.F.Y., Berthier, J., Enriquez, J., Hestroffer, D., Le Mignant, D., de Pater, I., 2006. Shape, size and multiplicity of main-belt asteroids. I. Keck adaptive optics survey. Icarus 185, 39–63. Matsui, T., Waza, T., Kani, K., Suzuki, S., 1982. Laboratory simulation of planetesimal collisions. J. Geophys. Res. 87, 10968–10982. Matsui, T., Waza, T., Kani, K., 1984. Destruction of rocks by low velocity impact and its implications for accretion and fragmentation processes of planetesimals. J. Geophys. Res. Suppl. 89, B700–B706. Michikami, T., and 11 colleagues, 2008. Size-frequency statistics of boulders on global surface of Asteroid 25143 Itokawa. Earth Planets Space 60, 13–20. Miyamoto, H., and 14 colleagues, 2007. Regolith migration and sorting on Asteroid Itokawa. Science 316, 1011–1014. Ostro, S.J., Giorgini, J.D., Benner, L.A.M., 2007. Radar reconnaissance of near-Earth asteroids. Proc. IAU Symp. 236, 143–150. Richardson, J.E., Melosh, H.J., Greenberg, R.J., O’Brien, D.P., 2005. The global effects of impact-induced seismic activity on fractured asteroid surface morphology. Icarus 179, 325–349. Slivan, S.M., 2002. Spin vector alignment of Koronis family asteroids. Nature 419, 49–51. Tanga, P., Hestroffer, D., Cellino, A., Lattanzi, M., Di Martino, M., Zappala, V., 2003. Asteroid observations with the Hubble Space Telescope. II. Duplicity search and size measurements for 6 asteroids. Astron. Astrophys. 401, 733–741. Thomas, P.C., Veverka, J., Robinson, M.S., Murchie, S., 2001. Shoemaker crater as the source of most ejecta blocks on the Asteroid 433 Eros. Nature 413, 394–396.
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The shape distribution of boulders on Asteroid 25143 Itokawa: Comparison with fragments from impact experiments Tatsuhiro Michikami a,b,*, Akiko M. Nakamura c, Naru Hirata d a
Fukushima National College of Technology, Iwaki, Fukushima 970-8034, Japan Planetary and Space Sciences Research Institute, The Open University, Walton Hall, Milton Keynes, MK7 6AA, United Kingdom c Graduate School of Science/Center for Planetary Science, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan d Department of Computer Science and Engineering/ARC-Space, CAIST, The University of Aizu, Ikki-machi, Aizu-Wakamatsu, Fukushima 965-8580, Japan b
a r t i c l e
i n f o
Article history: Received 10 June 2009 Revised 30 September 2009 Accepted 4 October 2009 Available online 22 October 2009 Keywords: Asteroids Asteroid Itokawa Asteroid Eros Asteroids, Surfaces Regoliths
a b s t r a c t Laboratory impact experiments have found that the shape of fragments over a broad size range is distribp uted around the mean value of the axial ratio 2: 2:1, which is independent of a wide range of experimental conditions. We report the shape statistics of boulders with size of 0.1–30 m on the surface of Asteroid 25143 Itokawa based on high-resolution images obtained by the Hayabusa spacecraft in order to investigate whether their shape distribution is similar to the distribution obtained for fragments (smaller than 0.1 m) in laboratory impact experiments. We also investigated the shapes of boulders with size of 0.1–150 m on Asteroid 433 Eros using a few arbitrary selected images by the NEAR spacecraft, in order to compare those with the shapes on Asteroid Itokawa. In addition, the shapes of small- and fastrotating asteroids (diameter <200 m and rotation period <1 h), which are natural fragments from past impact events among asteroids, were inferred from archived light curve data taken by ground-based telescopes. The results show that the shape distributions of laboratory fragments are similar to those of the boulders on Eros and of the small- and fast-rotating asteroids, but are different from those on Itokawa. However, we propose that the apparent difference between the boulders of Itokawa and the laboratory fragments is due to the migration of boulders. Therefore, we suggest that the shape distributions of the boulders ranging from 0.1 to 150 m in size and the small- and fast-rotating asteroids are similar to those obtained for the fragments generated in laboratory impact experiments. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Laboratory impact experiments have found that the shapes of fragments as defined by axes a, b and c, these being the maximum dimensions of the fragment in three mutually orthogonal planes (a P b P c) are distributed around mean values of the axial ratios b/a 0.7 and c/a 0.5, i.e., corresponding to a:b:c in the simple p proportion 2: 2:1. This result indicates a general property of collisional fragments which is repeated with great regularity in widely different experimental conditions such as projectile velocity, target shape, composition and strength (Fujiwara et al., 1978; Matsui et al., 1982, 1984; Bianchi et al., 1984; Capaccioni et al., 1984, 1986). In general, the shape distributions of small asteroids less than tens of kilometers in diameter are considered to be similar to distributions obtained for fragments generated in laboratory impact experiments (Fujiwara et al., 1978; Capaccioni et al., 1984, 1986). The shapes of most asteroids can be inferred from the ob-
* Corresponding author. Address: Fukushima National College of Technology, Iwaki, Fukushima 970-8034, Japan. E-mail address: [email protected] (T. Michikami). 0019-1035/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2009.10.008
served light curve amplitude. The observed amplitudes imply that the shape distribution of small asteroids is similar to those of laboratory fragments although the observed small asteroids tend to have slightly more elongated shapes (e.g., Catullo et al., 1984; Binzel et al., 1989). Most asteroids are fragments produced as a result of the collisional disruption of parent bodies, or are gravitationally bound rubble piles of fragments. The small asteroids have a wide range of shape elongations, while the shapes of large asteroids are more spherical due to self-gravitational effects (e.g., Capaccioni et al., 1984). The sizes of the observed laboratory fragments are less than 101 m, and the sizes of the observed asteroids are larger than 102 m. In previous studies, there has been no research on the shape distribution of the fragments with size of between 101 and 102 m. In the last few years, the high-resolution images taken by the NEAR and Hayabusa spacecraft have shown that there are many boulders on Asteroids Eros and Itokawa, respectively (e.g., Thomas et al., 2001; Fujiwara et al., 2006). The sizes of these boulders range from tens of centimeters to several tens of meters, occasionally as large as 150 meters. The boulders are formed by impact cratering or catastrophic disruption of the parent asteroid. The discovery of many
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boulders provides the opportunity for the statistical estimation of the shapes of the fragments in space. We report the shape statistics of boulders on the surface of Itokawa based on high-resolution images obtained by the Hayabusa spacecraft. Then we investigate, though limited to only a very small sampling, the boulder shapes on Asteroid Eros in order to compare those with the shapes of boulders on Asteroid Itokawa. In addition, in order to confirm the shapes of small asteroids as fragments with size less than 200 m, the shapes of fast-spinning asteroids (with spin period less than 1 h) are inferred from the archived light curve observation data (Harris and Pravec; EAR-A5-DDR-DERIVED-LIGHTCURVE-V10.0. ‘‘http://www.psi.edu/pds/ resource/lc.html”). In the last few years, smaller (less than 200 m), fast-rotating asteroids have been discovered and these are considered to be monolithic bodies (not rubble piles) generated by impact cratering or catastrophic disruption of the parent asteroids (Holsapple, 2007). Based on these analyses, we investigate whether the shape distributions of these fragments found in space ranging from tens of centimeters to a few hundred meters are similar to the distribution obtained for fragments less than 10 cm in diameter generated in laboratory impact experiments. 2. Methodology We define isolated positive relief features on the surface of an asteroid as boulders. Isolated positive relief features might include partially buried boulders, weathered boulders with subdued outlines, piles of regolithic scree, protruding bedrocks, raised crater rims and intersecting crater walls. From the images taken by spacecraft, it is difficult to distinguish clearly between these features, especially for small positive relief. For large positive relief features (>5 m), we took out only what is considered to be a boulder by comparing a few hundred images with stereo coverage acquired from the nominal hovering position (Home position) at an altitude of about 7 km. Almost all large positive relief features would be boulders. The shape statistics of the large boulders (>5 m) on the entire surface of Itokawa were investigated from eight images (resolutions 0.4 m/pixel) acquired on 19–26th October 2005. In this study, we call the mean size of a boulder larger than 5 m ‘‘a large boulder”. This critical value is not hampered by the limited spatial resolution on the entire asteroid imaged by the spacecraft. When we measured the size of a boulder, we chose a best image in which the outline of the boulder is clear, the emission angle between surface normal and spacecraft position is the smallest and the apparent dimension is the longest in the eight images. We started by measuring the apparent longest dimension of the boulder (a0 ) and then the apparent longest dimension (b0 ) perpendicular to this. We made corrections in the axial ratios of the large boulders on Itokawa because the apparent dimension of the boulder might be somewhat affected by the emission angle (data excluding the large boulders on Itokawa were not corrected because the emission angles of these data are small). In our study, as a first approximation, the correction value of the axial ratio is calculated using the following equation: 0
b b ¼ a a0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sin j þ cos2 j cos2 e 2
sin j cos2 e þ cos2 j
;
ð1Þ
where a and b are the correction value of a0 and b0 , respectively, and e and j are the emission angle and the angle between the asteroid surface projection axis of the emission vector and the a0 axis, respectively. This equation can be obtained geometrically assuming that the a0 and b0 are the projected axes of the a and b of the boulder on the plane with e and j (note that, in this equation, we do not take into account of the ellipsoid or the true shape of the boulder but
treat it as two axes a and b). If the direction of the a0 axis is the same as the asteroid surface projection axis of the emission vector (j = 0°), b0 equals b and a0 becomes a(cos e), eventually the correction value of the axial ratio (b/a) becomes (b0 /a0 )(cos e). If the direction of the a0 axis is perpendicular to the asteroid surface projection axis of the emission vector (j = 90°), a0 equals a and b0 becomes b(cos e), eventually the correction value of the axial ratio (b/a) becomes (b0 /a0 )(1/ cos e). When we make corrections, the correction values of the axial ratios in small fraction of boulders become over 1; the dimension of the a axis becomes smaller than that of the b axis. In this case, we excluded these data because the longest dimension cannot be measured correctly. Note that Eq. (1) is not applicable when the orientations of the apparent longest axis are not parallel to the asteroid surface. However, in our observation, because the orientations of the apparent dimension axis of most large boulders are parallel to the asteroid surface, Eq. (1) would be applicable in statistical viewpoint. The dimensions of a and b might not represent the actual longest and middle axes of the boulder because these are apparent dimensions of the boulder on the asteroid surface, which might be partially buried. The shape statistics of small boulders with size less than 5 m were derived from several smaller, representative regions’ images; these were estimated from six close-up images acquired on 9–12th November 2005 (Fig. 1). We counted all positive relief features in these six images and defined these as boulders. Boulder mapping is affected by variations in spatial resolution and in the availability of stereo coverage. The regions at middle and low latitudes were viewed from several images at resolution 0.4 m/pixel and with substantial stereo coverage. In contrast, the regions at high latitude were viewed from only a few images at resolution 0.4 m/pixel and with no stereo coverage. Therefore, it is possible that several boulders located at high latitude were overlooked in our counting.
3. Observation results of the boulders on Itokawa There are 373 boulders larger than 5 m on the entire surface of Itokawa, using the method of Michikami et al. (2008). The numbers of boulders larger than 10 m is about 40. The largest boulder is called ‘‘Yoshinodai” with dimensions of 50 m by 30 m by 20 m (Fujiwara et al., 2006). The shape distributions of large boulders on the surface of Itokawa and of the fragments in laboratory experiments (Capaccioni et al., 1984) are shown in Figs. 2 and 3, respectively. Note that the mean value of the apparent axial ratio (b/a) of the boulders larger than 5 m on the entire surface of Itokawa is 0.62 (Table 1), which is smaller than that (0.72) of the fragments in laboratory experiments. Both of rank-sum test and t test show that the difference between the observed large boulders and laboratory fragments is statistically significant at the 95% confidence level. The typical large boulders of Itokawa appear to have somewhat more elongated shapes compared with laboratory fragments. The apparent axial ratios of the large boulders are distributed with a gentle peak compared with those of the fragments in the laboratory experiments; these boulders have various shapes such as spheres and elongated ellipses. In particular, there are no fragments with b/a < 0.3 in laboratory experiments, while there are several large boulders with b/a < 0.3 on Itokawa. The mean value of the apparent axial ratio of the boulders larger than 10 m is also similar to that of the boulders larger than 5 m (Table 1). We also investigated the shape distributions of the large boulders among different areas of Itokawa, such as the east and west sides, the rough and smooth areas, and the ‘‘head” and ‘‘body” portions. The surface of Itokawa is separated into ‘‘rough area”, which occupies about 80% of Itokawa’s surface with numerous boulders,
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Fig. 1. (a) East side of Itokawa. (b–g) Close-up images acquired on 9–12th November 2005. Scale bars are indicated on the top right of each image. Image IDs: (a) ST2417413276, (b) ST2539437177, (c) ST2539429953, (d) ST2539423137, (e) ST2532629277, (f) ST2539451609 and (g) ST2539444467. The locations of the regions covered by images (b–g) are shown in (a).
and ‘‘smooth area” covered by finer particles. A surprising number of positive relief features appear on the rough area. The shape of Itokawa resembles a sea otter, with a smaller, rather round head and a larger oval body mass. The results show that the mean values of the apparent axial ratios among these areas are similar (Table 1). As an exception, the mean value (0.56) of the apparent axial ratio on the smooth area is smaller than those of other areas. The difference might be due to the small number of large boulders on the smooth area. As mentioned before, the shape distributions of the boulders less than 5 m were estimated from six close-up images acquired on 9–12th November 2005. These images were sampled from the boundary of MUSES-C Regio and the rough area of the larger body moving from the west to the east along the equator of the east side (Fig. 1). The altitude of the spacecraft ranged from 60 to 646 m. Fig. 4 shows the shape distributions of the boulders ranging from 0.1 to 5 m in six close-up images. The counted numbers of the
boulders are several hundreds for each image. Image (e) in Fig. 1 covers a wide area and contains three smaller areas of images (b–d). Note that the mean values of the apparent axial ratio (b/a) of all of the boulders with size of 0.1–5 m is 0.68 (Table 2), which is slightly smaller than that (0.72) of the fragments in laboratory experiments. Although the difference between these looks small, it is statistically significant at the 95% confidence level (both of rank-sum test and t test). The small boulders of Itokawa also appear to have slightly more elongated shapes compared with those of laboratory fragments. The mean values of the apparent axial ratio for each image appear to be roughly the same (Table 2). The peaks of the shape distributions differ slightly for each image (Fig. 4). The shape distributions of the boulders on images (d) and (f) are peaked at b/a 0.6, while other shape distributions are peaked at b/a 0.7. The mean values (0.68) of the apparent axial ratio of all of the boulders with size of 0.1–5 m is larger than that (0.62) of the boul-
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Fig. 2. The shape distributions of boulders larger than 5 m on the surface of Itokawa. The horizontal axis shows the apparent axial ratios (b/a) of the boulders. The vertical axis indicates the number of boulders. We made corrections in the axial ratios of the large boulders on Itokawa using Eq. (1). The total shape distributions with the correction and without the correction are shown on the top left and the top right, respectively. The total number of the large boulders is 373 and the total number of those with the correction value is 330. The mean value of the apparent axial ratio (b/a) of the large boulders is 0.62, which is similar to that without the correction (0.61). The surface is divided into different areas, such as the east and west sides, the rough and smooth areas, and the head and body portions. The mean values of apparent axial ratios in each area are given in Table 1. Data were collected from the eight images of ST2473604354, ST2481211873, ST2482160259, ST2484352917, ST2485860275, ST2492225173, ST2492513077, and ST2493031594 at an altitude between 3.779 and 4.913 km.
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and 0156087736). The mean apparent axial ratio of these boulders with size of 4–17 m is 0.72 (the counted number is 41), and this value is also similar to those of laboratory fragments. The calculations of rank-sum test and t test show that these distributions are also similar. Still smaller boulders with size of 0.1–3.8 m are assessed from final images of the landing site on Eros acquired by NEAR Shoemaker on February 12, 2001 (Image IDs 0157416873, 0157417133 and 0157417198). The total counted number is 163. The apparent mean axial ratio of small boulders with size of 0.1– 3.8 m on the close-up images of Eros is 0.73, and this value is also similar to those of laboratory fragments. The calculations of ranksum test and t test show that these distributions are also similar. Therefore, the shape distribution of the boulders in the wide ranges of sizes on Eros is thought to be similar to those of laboratory fragments, although these data are derived from a small number of sample images. Fig. 3. The shape distribution of fragments in laboratory experiments (Fig. 2B in Capaccioni et al. (1984)). The mean value of the axial ratio (b/a) of the fragments is 0.72.
Table 1 The mean values of axial ratios (b/a) of boulders larger than 5 m in each area of Itokawa. These values are the correction ones using Eq. (1).
Total (without correction) Total East West Head Body Rough Smooth >10 m
Mean
Standard deviation
Number of boulders
0.61
0.19
373
0.62 0.61 0.64 0.62 0.62 0.63 0.56 0.63
0.19 0.18 0.20 0.20 0.18 0.19 0.12 0.19
330 166 164 101 229 307 23 38
ders larger than 5 m on the entire surface of Itokawa (Table 2). The small boulders less than 5 m appear to have less elongated shapes compared with the boulders larger than 5 m on the entire surface. In image (e) which covers a wide area, the mean values of the axial ratio (b/a) of the boulders with size of 1–5 m and with size of 2– 5 m are 0.66 and 0.60, respectively. This means that the mean values of the apparent axial ratio (b/a) of the boulders decrease with increasing the size; as the size increases, it looks like the boulders’ shape is getting more elongated for the size range of 0.1–5 m. 4. Shape distribution of boulders on Eros In order to compare with the shape of the boulders on Itokawa, preliminary measurements of shape of the boulders on Eros have been made in a few arbitrary selected locations from six images acquired by the NEAR Shoemaker spacecraft. The shapes of the boulders on Eros were measured using the same method for investigating the shape of the boulders on Itokawa; the shape distributions are shown in Fig. 5. Firstly, the population of large boulders with size of 60–150 m on Eros is sampled from the image taken during December 25, 2000 (Image ID 0153130598). The image shows inside the rim of Eros saddle region and the orbital altitude is 38 km (resolution 3 m/pixel). The mean apparent axial ratio of large boulders with size of 60–150 m is 0.73 (the counted number is 20), and this value is similar to those of laboratory fragments. The calculations of rank-sum test and t test show that these distributions are similar. Secondly, the shapes of relatively smallsized boulders with size of 4–17 m are derived from two low-altitude images in January 26 and 28, 2001 (Image IDs 0155888661
5. Shape distribution of small- and fast-rotating asteroids According to Binzel et al. (1989), the observed mean amplitude of light curve of asteroids is slightly higher than expected for the shape distribution of laboratory fragments; the observed asteroids tend to have slightly more elongated shapes. There is as yet no clear evidence to show that the shape distribution of small asteroids is the same as those of fragments in laboratory impact experiments. Nonellipsoidal shapes, scattering properties of the surface, shadowing and albedo effects at nonzero phase angles may be enhancing the observed amplitudes. Space missions have visited only a few asteroids and shapes for these have been determined accurately (e.g., Fujiwara et al., 2006). Radar observations have revealed shapes of Near-Earth Objects encountering the Earth (e.g., Ostro et al., 2007), while high-resolution imaging is restricted to the largest asteroids (Tanga et al., 2003; Marchis et al., 2006). In order to confirm the shapes of small asteroids as fragments with size less than 200 m, we researched the shapes of the fastrotating (rotation period less than 1 h) asteroids. Fast-rotating asteroids have been discovered in the last few years. They have diameters of less than 200 m, and are all Near Earth Asteroids. These spin speeds clearly exceed the rubble-pile spin limits (Holsapple, 2007). Thus, these small- and fast-rotating asteroids must have some tensile strength and are considered to be monolithic bodies generated by impact cratering or catastrophic disruption of the parent asteroids (Holsapple, 2007). Therefore, it is likely that these small asteroids have the original shapes of the fragments produced by impact phenomena. Light curve amplitudes (defined as the peak-to-peak variation) provide a crude indicator for asteroid shapes. The compilation of light curve amplitude data is acquired from the asteroid complication of Harris and Pravec (EAR-A-5-DDR-DERIVED-LIGHTCURVEV10.0). According to Binzel et al. (1989), for a triaxial ellipsoid rotating about the c axis, the light curve amplitude is given by
AðhÞ ¼ 2:5 log
a b
2
1:25 log
a2 cos2 h þ c2 sin h 2
2
!
b cos2 h þ c2 sin h
;
ð2Þ
where h is the polar aspect viewing angle (the angle between the line of sight and the axis of rotation). If an asteroid is viewed pole-on (h = 0°), then no change in projected surface area is seen and the expected amplitude is zero. If an asteroid is viewed at an equatorial aspect (h = 90°) such that the rotation axis is perpendicular to the line of sight, then the second term in the above equation is zero and the maximum amplitude would be expected as the projected surface area varies from pac to pbc. The polar aspect viewing angle is unknown for an asteroid observed during only one opposition (Magnusson et al., 1989). If we assume an isotropic distribution of asteroid spin vectors, then
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Fig. 4. The shape distributions of boulders ranging from 0.1 to 5 m on the six close-up images (corresponding to Fig. 1). The mean values of apparent axial ratios in each image are given in Table 2.
T. Michikami et al. / Icarus 207 (2010) 277–284 Table 2 The mean values of axial ratios (b/a) of boulders ranging from 0.1 to 5 m in each close-up images of Itokawa. No.
Mean
Standard deviation
Number of boulders
Size range (m)
b c d e
0.67 0.67 0.66 0.68 0.66 0.60 0.69 0.69 0.68
0.16 0.17 0.16 0.16 0.17 0.17 0.16 0.15 0.17
276 410 439 495 309 53 173 240 2033
0.12–4.65 0.22–3.84 0.23–2.74 0.58–5.37 1–5.37 2–5.37 0.14–2.15 0.05–2.56 0.05–5.37
f g Total
the mean value of the aspect angles observed during one opposition will be 60° (Binzel et al., 1989). However, a recent study of the Koronis family shows that the distribution of asteroid spin vectors is not isotropic (Slivan, 2002). It is possible that the rotation axis of some asteroids can become perpendicular to the asteroid’s orbit plane. In this study, we calculated the axial ratio of small- and fast-rotating asteroids from the amplitudes of these asteroids using the value of h = 90°, for simplicity. If the distribution of the asteroid spin vectors is isotropic, then the axial ratio becomes small compared with our estimated value. The shape distribution of 42 small- and fast-rotating asteroids (diameter <200 m and rotation period <1 h) is shown in Fig. 6. Note that the mean axial ratio is 0.71, which is similar to those of laboratory fragments. The calculations of rank-sum test and t test show that these distributions are similar.
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6. Discussion The shape distributions of the boulders with size of 0.1–150 m on Eros and the small- and fast-rotating asteroids (diameter <200 m and rotation period <1 h) are similar to those of fragments generated in laboratory impact experiments. On the other hand, the shape distributions of boulders on Itokawa are different from those of laboratory fragments; the boulders on Itokawa appear to have more elongated shapes. However, the actual shape distribution of the boulders on Itokawa would be similar to those of laboratory fragments. The apparent difference between the boulders of Itokawa and the laboratory fragments would be due to the migration of boulders (regolith migration). The regolith migration indicated by Miyamoto et al. (2007) brings about the different sedimentation of the boulders between Itokawa and Eros. The explanation of regolith migration on Itokawa is as follows. Most of boulders on Itokawa originated from the disruption of a larger parent body, and then these boulders were uniformly spread over the entire surface (Miyamoto et al., 2007; Michikami et al., 2008). The smaller boulders (gravels) were redistributed after their accumulation by global vibration (seismic shaking) which was caused by impact cratering, etc. Finer particles such as gravel migrated into the smooth terrains which are areas of low gravitational potential. This is because finer particles have higher mobility due to their lower friction angle. On the other hand, the larger boulders could not move easily and some large boulders were stranded at the surface owing to their large friction angle and then form rough terrains. These mechanisms are explained in detail by Miyamoto et al. (2007) as granular pro-
Fig. 5. The shape distributions of the boulders of size 0.1–150 m on Eros, which are determined for a few arbitrary selected locations from six images acquired by NEAR Shoemaker. The mean apparent values of axial ratios are 0.73 (for size 60–150 m), 0.72 (for size 4–17 m) and 0.73 (for size 0.1–3.8 m).
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tory-scale collision experiments. However, we believe that the possibility of this is low, because the shape distribution of boulders in the same size range on Eros is similar to laboratory fragments. Finally, we suggest that the actual shape distributions of the boulders on Itokawa are similar to those of laboratory fragments. Based on our analyses, we conclude that the shape distributions of the boulders ranging from 0.1 to 150 m in size and the smalland fast-rotating asteroids (diameter <200 m and rotation period <1 h) are similar to that obtained for fragments generated in laboratory impact experiments. Acknowledgments We would like to thank all members of Hayabusa mission team for their support of the data acquisition. We thank S. Green and A. Hagermann for useful discussion. Fig. 6. The shape distribution of the small- and fast-rotating asteroids (diameter <200 m and rotation period <1 h). The mean value of axial rations is 0.71.
References
cesses (granular processes become major resurfacing processes because of Itokawa’s small size). The c axis of the gravels gradually would become perpendicular to the surface of Itokawa during the granular process, owing to their lower friction angle and gravitational stability. As a result, the apparent shape distributions of small boulders (gravels) tend to approach those of laboratory fragments as the size of small boulders decreases: For boulders with sizes ranging from 2 to 5 m, 1 to 5 m, and 0.1 to 5 m, the mean apparent axial ratios are 0.60, 0.66, and 0.68, respectively. On the other hand, the orientations of the c axis of some large boulders may not be perpendicular to the surface because it is likely that some large boulders are stranded at the surface and remain at their original orientations owing to their large friction angle (note that, in our observation, the orientations of the a axis of most large boulders are parallel to the asteroid surface. Only the orientations of the c axis (or the b axis) would be somewhat random). Consequently, the appearance of the large boulders tends to more elongated shapes: the apparent axial ratios of the boulders larger than 5 m is 0.62. As one piece of evidence, the very large boulder near the neck between head and body portions is taller than it is wide, which is a rather unstable orientation. On Eros, almost all boulders would be reoriented after their accumulation by global vibration (e.g., Richardson et al., 2005) because the gravity of Eros is considerably larger than that of Itokawa. Therefore, the c axis of all boulders are thought to become perpendicular to the surface, and then the shape distribution of the boulders on Eros is similar to those of laboratory fragments: the mean apparent axial ratio of the boulders on Eros is about 0.73. There is a possibility that the approximation of Eq. (1) is not enough to make corrections in the axial ratio of the boulders. However, the apparent mean axial ratio of the boulders which were observed at the emission angles near 0° is 0.63 (the counted number 32), which is similar to that of the total boulders over the entire surface of Itokawa. Therefore, the approximation of Eq. (1) is considered to be enough to make corrections in the axial ratio of the boulders. Note that we do not rule out the possibility that the boulders on Itokawa are actually more elongated than predicted by the labora-
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